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Math4200,HonorsPaper RyanWallentine,SpringSemester2013 “Godmadenaturalnumbers;allelseistheworkofman.” ‐‐LeopoldKronecker Theabovequotestatesthatthenaturalnumbersarewhatweweregiven,andallother numbers,suchasintegersandrationalnumbers,werecreatedfromthem.Thisimpliesthatall othersetsofnumberscanbeformedoutofthenaturalnumbers.Theobjectofthispaperistoshow theconnectionbetweeneachsetofnumbersbyshowinghowwecanformeachsetfromthe naturalnumbers. Wewillbeginbydefiningallourterms.LetJrepresentthesetofallnaturalnumbers,W representthesetofallwholenumbers,Zrepresentthesetofallintegers,Qrepresentthesetofall rationalnumbers,andRrepresentthesetofallrealnumbers.WewillusePeano’sAxiomstodefine thenaturalnumbersinordertogiveusastartingpoint.Peano’sAxiomsareasfollows: Axiom1:1isanaturalnumber.Thatis,oursetisnotempty;itcontainsanobjectcalled “one.” Axiom2:Foreachx,thereexistsexactlyonenaturalnumber,calledthesuccessorofx, whichwillbedenotedbyx’. thinkx’ x 1 Axiom3:Wealwayshavex’ 1.Thatis,thereisnonumberwhosesuccessoris1. Axiom4:Ifx’ y’thenx y.Thatis,foranygivennumber,thereexistseithernonumberor exactlyonenumberwhosesuccessoristhegivennumber. Axiom5 AxiomofInduction :LettherebegivenasetJofnaturalnumbers,withthe followingproperties: I. II. J 1belongstoJ. IfxbelongstoJ,thensodoesx’. ThenJcontainsallthenaturalnumbers.Thus,if1’ 2,2’ 3,3’ 4,…,then 1,2,3,4,… . Fromthisdefinitionofthenaturalnumbers,wecaneasilycreatethesetofallwhole numbersbyincludingzero.WewillletWhavethesamepropertiesasJ,exceptforwewilllet0 belongtoWsuchthat0’ 1and0isnotthesuccessorofanynumber.Thus,W 0,1,2,3,… .We willnowdefineadditionandmultiplicationasfollows: Addition :Forallx,x 1 x’.Forallxandy,x y’ x y ’. Noticethat2 1’,sowehavethefollowing:4 2 4 1’ 4 1 ’ 4’ ’ 6. Multiplication * :Forallx,x*1 x.Forallxandy,x*y’ x*y x. Forexample,ify 1andx 4,thenx*y’ 4*1’ 4*1 4 4 4 8. Wewillnowdefine0asfollows:0 x xandx*0 0forallx.Fromherewecaneasilycreate theintegersbyletting–xdenotetheadditiveinverseofx.Thismeansthatx ‐x 0forallx. Theintegers,willthenbedefinedasfollows: Z x:x∈J ∪ 0 ∪ ‐x:x∈J ThusZ …,‐2,‐1,0,1,2,… . Toseemoreaboutbuildingtheintegers,see http://en.wikipedia.org/wiki/Integer#Construction . Wewillnowconstructtherationalnumbersfromtheintegersbyconstructingequivalence classesusingtheintegers.Theseequivalenceclasseswillbepairsofintegersputtogetheras follows: a,b :a,b∈Z,b 0 .Also, a,b isequivalentto c,d ifandonlyifad bc think a,b and c,d as and respectively. Forexample,let 3,7 denotetheequivalencedefinedinthepair 3,7 .Then, 3,7 a,b ∈S: a,b 3,7 .Forexample, 6,14 3,7 .Wecanrepresent 3,7 bythepairs 3,7 or 6,14 ,ormoreformerlyas or .ThesetofallrationalnumbersQis builtfromtheseequivalenceclasses. Furtherinformationaboutconstructingtherationals: http://en.wikipedia.org/wiki/Rational_number#Formal_construction .Additionand multiplicationfortherationalnumbersareasfollows: Addition: a,b c,d ad bc,bd Multiplication: a,b * c,d ac,bd Nowwearereadytoconstructtherealnumbers.LetT rn :rnisrationaland rn isa Cauchysequence. Wewilllettwosequences rn and sn inTbeequivalenttoeachotherifand onlyiflim → 0.Fromhere,itiseasytoshowthatthisrelationisreflexive,transitive, andsymmetric.ThesetTispartitionedintodisjointequivalenceclasseswhichwewillcallreal numbers.Forexample,letan 2foralln.Theequivalenceclass an istherealnumber2.The realnumber2isalsorepresentedby 2 .Sincelim lim → 2 2 2 2 → 2 2andlim → 2 2,then 0sincethelimitofthesumisthesumofthelimits. Considertherealnumbere.Wecanrepresentthisnumberbythelimitofthebinomial expansion:lim → 1 1 .Considerthebinomialexpansionforn 1,000,000.Wethenhave 2.71828046…whichweseeisveryclosetothevalueofe.Anotherwayto approximatethenumbereisthroughtheTaylorseriesexpansionfor givesus∑ ! ! ! ! ! ∑ ! forx 1.This ⋯Byaddingthetermsofthisseriesthroughn 15,weget 2.71828182…whichisalsoratherclose.Thesecontinuetogetcloserandclosertoeasn approachesinfinity. Nowconsidertherealnumber .Justlikee,therearevariouswaysthatwecan approximate aswell.Forexample,wehavetheGregory‐Leibnizseriesasfollows: 4∑ 4 ⋯ .Thisapproximation,however,convergestooslowlyforit tobeapracticalapproximation.AnotherapproximationwasgivenbyNewton: ! 2∑ 2 1 ! 1 1 1 ⋯ . Wewillnowlookataseriestorepresent√2.Letrnbethesequenceoflargestrational numberswithdenominatorlessthanorequaltonsuchthat rn 2 2.Thisgivesusthefollowing sequence: 1,1, , , ,… whichisaCauchysequencesuchthatlim andsin x ∑ Thus,√2 ! ! showthat√2 ∑ ,thenwegetthat√2 2sin ! 2∑ √2.Giventhatsin ! √ ∑ . ! ⋯.Inalikemanner,byusingcos x inplaceofsin x ,wecan ! ! → 2 ! ! ! ! ⋯.Thusweseethatwecanfind variousrepresentationsforvariousrealnumbersincludingtheirrationalnumberslike ,e,and√2. WewillnowtakealookatthenumberΦ phi whichrepresentsthegoldenratio.This numberisinterestingbecausemanyartistsandarchitectshaveproportionedtheirworksto approximatetheratiobecausetheproportionisaestheticallypleasing.Twoquantitiesaandbare saidtobeinthegoldenratioΦif Φ.ThisresultsinΦ √ moreinformationcanbe foundathttp://en.wikipedia.org/wiki/Golden_ratio .Themostinterestingthingaboutthis numbercomesfromonemethodofitsconstruction.FromtheFibonaccisequencedefinedas fn fn‐1 fn‐2wheref0 0andf1 1,wecancreateasequencethatconvergestoΦ.Consider lim .Oneclosedformulaforfn → √ √ .Itiseasytoshowfromthisclosedformulathat Φ.Thus,thesequenceoftheratiosofsuccessivepairsofnumberstakenfromthe FibonaccisequencewillconvergetoΦ. √